LMFDB - Embedded newform 1425.2.c.j.799.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1425,2,Mod(799,1425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,-4,0,0,-4,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.j.799.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -3.41421i q^{7} +4.41421i q^{8} -1.00000 q^{9} -1.41421 q^{11} +3.82843i q^{12} +2.58579i q^{13} -8.24264 q^{14} +3.00000 q^{16} +6.82843i q^{17} +2.41421i q^{18} -1.00000 q^{19} -3.41421 q^{21} +3.41421i q^{22} -3.65685i q^{23} +4.41421 q^{24} +6.24264 q^{26} +1.00000i q^{27} +13.0711i q^{28} -5.07107 q^{29} -10.4853 q^{31} +1.58579i q^{32} +1.41421i q^{33} +16.4853 q^{34} +3.82843 q^{36} +3.07107i q^{37} +2.41421i q^{38} +2.58579 q^{39} -4.58579 q^{41} +8.24264i q^{42} +3.41421i q^{43} +5.41421 q^{44} -8.82843 q^{46} -11.6569i q^{47} -3.00000i q^{48} -4.65685 q^{49} +6.82843 q^{51} -9.89949i q^{52} +4.00000i q^{53} +2.41421 q^{54} +15.0711 q^{56} +1.00000i q^{57} +12.2426i q^{58} +8.48528 q^{59} -5.65685 q^{61} +25.3137i q^{62} +3.41421i q^{63} +9.82843 q^{64} +3.41421 q^{66} -12.0000i q^{67} -26.1421i q^{68} -3.65685 q^{69} +12.4853 q^{71} -4.41421i q^{72} -2.00000i q^{73} +7.41421 q^{74} +3.82843 q^{76} +4.82843i q^{77} -6.24264i q^{78} -11.3137 q^{79} +1.00000 q^{81} +11.0711i q^{82} +6.48528i q^{83} +13.0711 q^{84} +8.24264 q^{86} +5.07107i q^{87} -6.24264i q^{88} +14.7279 q^{89} +8.82843 q^{91} +14.0000i q^{92} +10.4853i q^{93} -28.1421 q^{94} +1.58579 q^{96} -4.24264i q^{97} +11.2426i q^{98} +1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 16 q^{14} + 12 q^{16} - 4 q^{19} - 8 q^{21} + 12 q^{24} + 8 q^{26} + 8 q^{29} - 8 q^{31} + 32 q^{34} + 4 q^{36} + 16 q^{39} - 24 q^{41} + 16 q^{44} - 24 q^{46} + 4 q^{49}+ \cdots + 12 q^{96}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 16 * q^14 + 12 * q^16 - 4 * q^19 - 8 * q^21 + 12 * q^24 + 8 * q^26 + 8 * q^29 - 8 * q^31 + 32 * q^34 + 4 * q^36 + 16 * q^39 - 24 * q^41 + 16 * q^44 - 24 * q^46 + 4 * q^49 + 16 * q^51 + 4 * q^54 + 32 * q^56 + 28 * q^64 + 8 * q^66 + 8 * q^69 + 16 * q^71 + 24 * q^74 + 4 * q^76 + 4 * q^81 + 24 * q^84 + 16 * q^86 + 8 * q^89 + 24 * q^91 - 56 * q^94 + 12 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.41421i	− 1.70711i	−0.521005	−	0.853553i	$-0.674443\pi$
$2$	− 2.41421i	− 1.70711i	0.521005	−	0.853553i	$-0.325557\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	−2.41421	−0.985599
$7$	− 3.41421i	− 1.29045i	−0.763992	−	0.645226i	$-0.776763\pi$
$7$	− 3.41421i	− 1.29045i	0.763992	−	0.645226i	$-0.223237\pi$
$8$	4.41421i	1.56066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	3.82843i	1.10517i
$13$	2.58579i	0.717168i	0.933497	+	0.358584i	$0.116740\pi$
$13$	2.58579i	0.717168i	−0.933497	+	0.358584i	$0.883260\pi$
$14$	−8.24264	−2.20294
$15$	0	0
$16$	3.00000	0.750000
$17$	6.82843i	1.65614i	0.560627	+	0.828068i	$0.310560\pi$
$17$	6.82843i	1.65614i	−0.560627	+	0.828068i	$0.689440\pi$
$18$	2.41421i	0.569036i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.41421	−0.745042
$22$	3.41421i	0.727913i
$23$	− 3.65685i	− 0.762507i	−0.924471	−	0.381253i	$-0.875493\pi$
$23$	− 3.65685i	− 0.762507i	0.924471	−	0.381253i	$-0.124507\pi$
$24$	4.41421	0.901048
$25$	0	0
$26$	6.24264	1.22428
$27$	1.00000i	0.192450i
$28$	13.0711i	2.47020i
$29$	−5.07107	−0.941674	−0.470837	−	0.882220i	$-0.656048\pi$
$29$	−5.07107	−0.941674	−0.470837	+	0.882220i	$0.656048\pi$
$30$	0	0
$31$	−10.4853	−1.88321	−0.941606	−	0.336717i	$-0.890684\pi$
$31$	−10.4853	−1.88321	−0.941606	+	0.336717i	$0.890684\pi$
$32$	1.58579i	0.280330i
$33$	1.41421i	0.246183i
$34$	16.4853	2.82720
$35$	0	0
$36$	3.82843	0.638071
$37$	3.07107i	0.504880i	0.967612	+	0.252440i	$0.0812331\pi$
$37$	3.07107i	0.504880i	−0.967612	+	0.252440i	$0.918767\pi$
$38$	2.41421i	0.391637i
$39$	2.58579	0.414057
$40$	0	0
$41$	−4.58579	−0.716180	−0.358090	−	0.933687i	$-0.616572\pi$
$41$	−4.58579	−0.716180	−0.358090	+	0.933687i	$0.616572\pi$
$42$	8.24264i	1.27187i
$43$	3.41421i	0.520663i	0.965519	+	0.260331i	$0.0838318\pi$
$43$	3.41421i	0.520663i	−0.965519	+	0.260331i	$0.916168\pi$
$44$	5.41421	0.816223
$45$	0	0
$46$	−8.82843	−1.30168
$47$	− 11.6569i	− 1.70033i	−0.526519	−	0.850163i	$-0.676503\pi$
$47$	− 11.6569i	− 1.70033i	0.526519	−	0.850163i	$-0.323497\pi$
$48$	− 3.00000i	− 0.433013i
$49$	−4.65685	−0.665265
$50$	0	0
$51$	6.82843	0.956171
$52$	− 9.89949i	− 1.37281i
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	2.41421	0.328533
$55$	0	0
$56$	15.0711	2.01396
$57$	1.00000i	0.132453i
$58$	12.2426i	1.60754i
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	25.3137i	3.21484i
$63$	3.41421i	0.430150i
$64$	9.82843	1.22855
$65$	0	0
$66$	3.41421	0.420261
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 26.1421i	− 3.17020i
$69$	−3.65685	−0.440234
$70$	0	0
$71$	12.4853	1.48173	0.740865	−	0.671654i	$-0.234416\pi$
$71$	12.4853	1.48173	0.740865	+	0.671654i	$0.234416\pi$
$72$	− 4.41421i	− 0.520220i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	7.41421	0.861885
$75$	0	0
$76$	3.82843	0.439151
$77$	4.82843i	0.550250i
$78$	− 6.24264i	− 0.706840i
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.0711i	1.22259i
$83$	6.48528i	0.711852i	0.934514	+	0.355926i	$0.115835\pi$
$83$	6.48528i	0.711852i	−0.934514	+	0.355926i	$0.884165\pi$
$84$	13.0711	1.42617
$85$	0	0
$86$	8.24264	0.888827
$87$	5.07107i	0.543676i
$88$	− 6.24264i	− 0.665468i
$89$	14.7279	1.56116	0.780578	−	0.625058i	$-0.214925\pi$
$89$	14.7279	1.56116	0.780578	+	0.625058i	$0.214925\pi$
$90$	0	0
$91$	8.82843	0.925471
$92$	14.0000i	1.45960i
$93$	10.4853i	1.08727i
$94$	−28.1421	−2.90264
$95$	0	0
$96$	1.58579	0.161849
$97$	− 4.24264i	− 0.430775i	−0.976529	−	0.215387i	$-0.930899\pi$
$97$	− 4.24264i	− 0.430775i	0.976529	−	0.215387i	$-0.0691014\pi$
$98$	11.2426i	1.13568i
$99$	1.41421	0.142134
$100$	0	0
$101$	0.828427	0.0824316	0.0412158	−	0.999150i	$-0.486877\pi$
$101$	0.828427	0.0824316	0.0412158	+	0.999150i	$0.486877\pi$
$102$	− 16.4853i	− 1.63229i
$103$	9.65685i	0.951518i	0.879576	+	0.475759i	$0.157827\pi$
$103$	9.65685i	0.951518i	−0.879576	+	0.475759i	$0.842173\pi$
$104$	−11.4142	−1.11926
$105$	0	0
$106$	9.65685	0.937957
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	− 3.82843i	− 0.368391i
$109$	−8.82843	−0.845610	−0.422805	−	0.906221i	$-0.638954\pi$
$109$	−8.82843	−0.845610	−0.422805	+	0.906221i	$0.638954\pi$
$110$	0	0
$111$	3.07107	0.291493
$112$	− 10.2426i	− 0.967839i
$113$	− 4.48528i	− 0.421940i	−0.977493	−	0.210970i	$-0.932338\pi$
$113$	− 4.48528i	− 0.421940i	0.977493	−	0.210970i	$-0.0676622\pi$
$114$	2.41421	0.226112
$115$	0	0
$116$	19.4142	1.80256
$117$	− 2.58579i	− 0.239056i
$118$	− 20.4853i	− 1.88582i
$119$	23.3137	2.13716
$120$	0	0
$121$	−9.00000	−0.818182
$122$	13.6569i	1.23643i
$123$	4.58579i	0.413486i
$124$	40.1421	3.60487
$125$	0	0
$126$	8.24264	0.734313
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 20.5563i	− 1.81694i
$129$	3.41421	0.300605
$130$	0	0
$131$	−8.72792	−0.762562	−0.381281	−	0.924459i	$-0.624517\pi$
$131$	−8.72792	−0.762562	−0.381281	+	0.924459i	$0.624517\pi$
$132$	− 5.41421i	− 0.471247i
$133$	3.41421i	0.296050i
$134$	−28.9706	−2.50268
$135$	0	0
$136$	−30.1421	−2.58467
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	8.82843i	0.751526i
$139$	−6.82843	−0.579180	−0.289590	−	0.957151i	$-0.593519\pi$
$139$	−6.82843	−0.579180	−0.289590	+	0.957151i	$0.593519\pi$
$140$	0	0
$141$	−11.6569	−0.981684
$142$	− 30.1421i	− 2.52947i
$143$	− 3.65685i	− 0.305802i
$144$	−3.00000	−0.250000
$145$	0	0
$146$	−4.82843	−0.399603
$147$	4.65685i	0.384091i
$148$	− 11.7574i	− 0.966449i
$149$	−7.65685	−0.627274	−0.313637	−	0.949543i	$-0.601547\pi$
$149$	−7.65685	−0.627274	−0.313637	+	0.949543i	$0.601547\pi$
$150$	0	0
$151$	−21.7990	−1.77398	−0.886988	−	0.461792i	$-0.847207\pi$
$151$	−21.7990	−1.77398	−0.886988	+	0.461792i	$0.847207\pi$
$152$	− 4.41421i	− 0.358040i
$153$	− 6.82843i	− 0.552046i
$154$	11.6569	0.939336
$155$	0	0
$156$	−9.89949	−0.792594
$157$	− 17.7990i	− 1.42051i	−0.703942	−	0.710257i	$-0.748579\pi$
$157$	− 17.7990i	− 1.42051i	0.703942	−	0.710257i	$-0.251421\pi$
$158$	27.3137i	2.17296i
$159$	4.00000	0.317221
$160$	0	0
$161$	−12.4853	−0.983978
$162$	− 2.41421i	− 0.189679i
$163$	− 11.8995i	− 0.932040i	−0.884774	−	0.466020i	$-0.845687\pi$
$163$	− 11.8995i	− 0.932040i	0.884774	−	0.466020i	$-0.154313\pi$
$164$	17.5563	1.37092
$165$	0	0
$166$	15.6569	1.21521
$167$	− 10.0000i	− 0.773823i	−0.922117	−	0.386912i	$-0.873542\pi$
$167$	− 10.0000i	− 0.773823i	0.922117	−	0.386912i	$-0.126458\pi$
$168$	− 15.0711i	− 1.16276i
$169$	6.31371	0.485670
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 13.0711i	− 0.996660i
$173$	22.1421i	1.68344i	0.539918	+	0.841718i	$0.318455\pi$
$173$	22.1421i	1.68344i	−0.539918	+	0.841718i	$0.681545\pi$
$174$	12.2426	0.928112
$175$	0	0
$176$	−4.24264	−0.319801
$177$	− 8.48528i	− 0.637793i
$178$	− 35.5563i	− 2.66506i
$179$	−22.8284	−1.70628	−0.853138	−	0.521685i	$-0.825304\pi$
$179$	−22.8284	−1.70628	−0.853138	+	0.521685i	$0.825304\pi$
$180$	0	0
$181$	−24.8284	−1.84548	−0.922741	−	0.385420i	$-0.874057\pi$
$181$	−24.8284	−1.84548	−0.922741	+	0.385420i	$0.874057\pi$
$182$	− 21.3137i	− 1.57988i
$183$	5.65685i	0.418167i
$184$	16.1421	1.19001
$185$	0	0
$186$	25.3137	1.85609
$187$	− 9.65685i	− 0.706179i
$188$	44.6274i	3.25479i
$189$	3.41421	0.248347
$190$	0	0
$191$	−17.8995	−1.29516	−0.647581	−	0.761997i	$-0.724219\pi$
$191$	−17.8995	−1.29516	−0.647581	+	0.761997i	$0.724219\pi$
$192$	− 9.82843i	− 0.709306i
$193$	− 0.928932i	− 0.0668660i	−0.999441	−	0.0334330i	$-0.989356\pi$
$193$	− 0.928932i	− 0.0668660i	0.999441	−	0.0334330i	$-0.0106440\pi$
$194$	−10.2426	−0.735379
$195$	0	0
$196$	17.8284	1.27346
$197$	9.17157i	0.653448i	0.945120	+	0.326724i	$0.105945\pi$
$197$	9.17157i	0.653448i	−0.945120	+	0.326724i	$0.894055\pi$
$198$	− 3.41421i	− 0.242638i
$199$	−0.485281	−0.0344007	−0.0172003	−	0.999852i	$-0.505475\pi$
$199$	−0.485281	−0.0344007	−0.0172003	+	0.999852i	$0.505475\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	− 2.00000i	− 0.140720i
$203$	17.3137i	1.21518i
$204$	−26.1421	−1.83032
$205$	0	0
$206$	23.3137	1.62434
$207$	3.65685i	0.254169i
$208$	7.75736i	0.537876i
$209$	1.41421	0.0978232
$210$	0	0
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	− 15.3137i	− 1.05175i
$213$	− 12.4853i	− 0.855477i
$214$	−19.3137	−1.32026
$215$	0	0
$216$	−4.41421	−0.300349
$217$	35.7990i	2.43019i
$218$	21.3137i	1.44355i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−17.6569	−1.18773
$222$	− 7.41421i	− 0.497609i
$223$	17.6569i	1.18239i	0.806529	+	0.591195i	$0.201344\pi$
$223$	17.6569i	1.18239i	−0.806529	+	0.591195i	$0.798656\pi$
$224$	5.41421	0.361752
$225$	0	0
$226$	−10.8284	−0.720296
$227$	− 14.9706i	− 0.993631i	−0.867856	−	0.496816i	$-0.834503\pi$
$227$	− 14.9706i	− 0.993631i	0.867856	−	0.496816i	$-0.165497\pi$
$228$	− 3.82843i	− 0.253544i
$229$	−9.65685	−0.638143	−0.319071	−	0.947731i	$-0.603371\pi$
$229$	−9.65685	−0.638143	−0.319071	+	0.947731i	$0.603371\pi$
$230$	0	0
$231$	4.82843	0.317687
$232$	− 22.3848i	− 1.46963i
$233$	− 19.6569i	− 1.28776i	−0.765125	−	0.643882i	$-0.777323\pi$
$233$	− 19.6569i	− 1.28776i	0.765125	−	0.643882i	$-0.222677\pi$
$234$	−6.24264	−0.408094
$235$	0	0
$236$	−32.4853	−2.11461
$237$	11.3137i	0.734904i
$238$	− 56.2843i	− 3.64837i
$239$	−5.41421	−0.350216	−0.175108	−	0.984549i	$-0.556028\pi$
$239$	−5.41421	−0.350216	−0.175108	+	0.984549i	$0.556028\pi$
$240$	0	0
$241$	18.9706	1.22200	0.611001	−	0.791630i	$-0.290767\pi$
$241$	18.9706	1.22200	0.611001	+	0.791630i	$0.290767\pi$
$242$	21.7279i	1.39672i
$243$	− 1.00000i	− 0.0641500i
$244$	21.6569	1.38644
$245$	0	0
$246$	11.0711	0.705866
$247$	− 2.58579i	− 0.164530i
$248$	− 46.2843i	− 2.93905i
$249$	6.48528	0.410988
$250$	0	0
$251$	27.0711	1.70871	0.854355	−	0.519689i	$-0.173952\pi$
$251$	27.0711	1.70871	0.854355	+	0.519689i	$0.173952\pi$
$252$	− 13.0711i	− 0.823400i
$253$	5.17157i	0.325134i
$254$	−19.3137	−1.21185
$255$	0	0
$256$	−29.9706	−1.87316
$257$	− 6.82843i	− 0.425946i	−0.977058	−	0.212973i	$-0.931685\pi$
$257$	− 6.82843i	− 0.425946i	0.977058	−	0.212973i	$-0.0683146\pi$
$258$	− 8.24264i	− 0.513164i
$259$	10.4853	0.651524
$260$	0	0
$261$	5.07107	0.313891
$262$	21.0711i	1.30177i
$263$	− 3.85786i	− 0.237886i	−0.992901	−	0.118943i	$-0.962049\pi$
$263$	− 3.85786i	− 0.237886i	0.992901	−	0.118943i	$-0.0379506\pi$
$264$	−6.24264	−0.384208
$265$	0	0
$266$	8.24264	0.505389
$267$	− 14.7279i	− 0.901334i
$268$	45.9411i	2.80630i
$269$	28.3848	1.73065	0.865325	−	0.501211i	$-0.167112\pi$
$269$	28.3848	1.73065	0.865325	+	0.501211i	$0.167112\pi$
$270$	0	0
$271$	−25.1716	−1.52906	−0.764532	−	0.644586i	$-0.777030\pi$
$271$	−25.1716	−1.52906	−0.764532	+	0.644586i	$0.777030\pi$
$272$	20.4853i	1.24210i
$273$	− 8.82843i	− 0.534321i
$274$	33.7990	2.04187
$275$	0	0
$276$	14.0000	0.842701
$277$	22.9706i	1.38017i	0.723730	+	0.690084i	$0.242426\pi$
$277$	22.9706i	1.38017i	−0.723730	+	0.690084i	$0.757574\pi$
$278$	16.4853i	0.988721i
$279$	10.4853	0.627737
$280$	0	0
$281$	10.7279	0.639974	0.319987	−	0.947422i	$-0.396321\pi$
$281$	10.7279	0.639974	0.319987	+	0.947422i	$0.396321\pi$
$282$	28.1421i	1.67584i
$283$	− 2.24264i	− 0.133311i	−0.997776	−	0.0666556i	$-0.978767\pi$
$283$	− 2.24264i	− 0.133311i	0.997776	−	0.0666556i	$-0.0212329\pi$
$284$	−47.7990	−2.83635
$285$	0	0
$286$	−8.82843	−0.522036
$287$	15.6569i	0.924195i
$288$	− 1.58579i	− 0.0934434i
$289$	−29.6274	−1.74279
$290$	0	0
$291$	−4.24264	−0.248708
$292$	7.65685i	0.448084i
$293$	7.79899i	0.455622i	0.973705	+	0.227811i	$0.0731568\pi$
$293$	7.79899i	0.455622i	−0.973705	+	0.227811i	$0.926843\pi$
$294$	11.2426	0.655684
$295$	0	0
$296$	−13.5563	−0.787947
$297$	− 1.41421i	− 0.0820610i
$298$	18.4853i	1.07082i
$299$	9.45584	0.546846
$300$	0	0
$301$	11.6569	0.671890
$302$	52.6274i	3.02837i
$303$	− 0.828427i	− 0.0475919i
$304$	−3.00000	−0.172062
$305$	0	0
$306$	−16.4853	−0.942401
$307$	31.7990i	1.81486i	0.420199	+	0.907432i	$0.361960\pi$
$307$	31.7990i	1.81486i	−0.420199	+	0.907432i	$0.638040\pi$
$308$	− 18.4853i	− 1.05330i
$309$	9.65685	0.549359
$310$	0	0
$311$	−23.7574	−1.34716	−0.673578	−	0.739116i	$-0.735244\pi$
$311$	−23.7574	−1.34716	−0.673578	+	0.739116i	$0.735244\pi$
$312$	11.4142i	0.646203i
$313$	− 26.4853i	− 1.49704i	−0.663114	−	0.748518i	$-0.730766\pi$
$313$	− 26.4853i	− 1.49704i	0.663114	−	0.748518i	$-0.269234\pi$
$314$	−42.9706	−2.42497
$315$	0	0
$316$	43.3137	2.43659
$317$	11.3137i	0.635441i	0.948184	+	0.317721i	$0.102917\pi$
$317$	11.3137i	0.635441i	−0.948184	+	0.317721i	$0.897083\pi$
$318$	− 9.65685i	− 0.541529i
$319$	7.17157	0.401531
$320$	0	0
$321$	−8.00000	−0.446516
$322$	30.1421i	1.67976i
$323$	− 6.82843i	− 0.379944i
$324$	−3.82843	−0.212690
$325$	0	0
$326$	−28.7279	−1.59109
$327$	8.82843i	0.488213i
$328$	− 20.2426i	− 1.11771i
$329$	−39.7990	−2.19419
$330$	0	0
$331$	12.8284	0.705114	0.352557	−	0.935790i	$-0.385312\pi$
$331$	12.8284	0.705114	0.352557	+	0.935790i	$0.385312\pi$
$332$	− 24.8284i	− 1.36264i
$333$	− 3.07107i	− 0.168293i
$334$	−24.1421	−1.32100
$335$	0	0
$336$	−10.2426	−0.558782
$337$	− 19.7574i	− 1.07625i	−0.842864	−	0.538126i	$-0.819132\pi$
$337$	− 19.7574i	− 1.07625i	0.842864	−	0.538126i	$-0.180868\pi$
$338$	− 15.2426i	− 0.829090i
$339$	−4.48528	−0.243607
$340$	0	0
$341$	14.8284	0.803004
$342$	− 2.41421i	− 0.130546i
$343$	− 8.00000i	− 0.431959i
$344$	−15.0711	−0.812578
$345$	0	0
$346$	53.4558	2.87380
$347$	6.48528i	0.348148i	0.984733	+	0.174074i	$0.0556932\pi$
$347$	6.48528i	0.348148i	−0.984733	+	0.174074i	$0.944307\pi$
$348$	− 19.4142i	− 1.04071i
$349$	−6.68629	−0.357909	−0.178954	−	0.983857i	$-0.557271\pi$
$349$	−6.68629	−0.357909	−0.178954	+	0.983857i	$0.557271\pi$
$350$	0	0
$351$	−2.58579	−0.138019
$352$	− 2.24264i	− 0.119533i
$353$	− 7.65685i	− 0.407533i	−0.979019	−	0.203767i	$-0.934682\pi$
$353$	− 7.65685i	− 0.407533i	0.979019	−	0.203767i	$-0.0653184\pi$
$354$	−20.4853	−1.08878
$355$	0	0
$356$	−56.3848	−2.98839
$357$	− 23.3137i	− 1.23389i
$358$	55.1127i	2.91280i
$359$	9.89949	0.522475	0.261238	−	0.965275i	$-0.415869\pi$
$359$	9.89949	0.522475	0.261238	+	0.965275i	$0.415869\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	59.9411i	3.15044i
$363$	9.00000i	0.472377i
$364$	−33.7990	−1.77155
$365$	0	0
$366$	13.6569	0.713855
$367$	16.5858i	0.865771i	0.901449	+	0.432886i	$0.142505\pi$
$367$	16.5858i	0.865771i	−0.901449	+	0.432886i	$0.857495\pi$
$368$	− 10.9706i	− 0.571880i
$369$	4.58579	0.238727
$370$	0	0
$371$	13.6569	0.709029
$372$	− 40.1421i	− 2.08127i
$373$	9.89949i	0.512576i	0.966600	+	0.256288i	$0.0824996\pi$
$373$	9.89949i	0.512576i	−0.966600	+	0.256288i	$0.917500\pi$
$374$	−23.3137	−1.20552
$375$	0	0
$376$	51.4558	2.65363
$377$	− 13.1127i	− 0.675338i
$378$	− 8.24264i	− 0.423956i
$379$	4.14214	0.212767	0.106384	−	0.994325i	$-0.466073\pi$
$379$	4.14214	0.212767	0.106384	+	0.994325i	$0.466073\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	43.2132i	2.21098i
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	−20.5563	−1.04901
$385$	0	0
$386$	−2.24264	−0.114147
$387$	− 3.41421i	− 0.173554i
$388$	16.2426i	0.824595i
$389$	18.9706	0.961846	0.480923	−	0.876763i	$-0.340302\pi$
$389$	18.9706	0.961846	0.480923	+	0.876763i	$0.340302\pi$
$390$	0	0
$391$	24.9706	1.26282
$392$	− 20.5563i	− 1.03825i
$393$	8.72792i	0.440265i
$394$	22.1421	1.11550
$395$	0	0
$396$	−5.41421	−0.272074
$397$	− 24.3431i	− 1.22175i	−0.791728	−	0.610874i	$-0.790818\pi$
$397$	− 24.3431i	− 1.22175i	0.791728	−	0.610874i	$-0.209182\pi$
$398$	1.17157i	0.0587256i
$399$	3.41421	0.170924
$400$	0	0
$401$	−18.0416	−0.900956	−0.450478	−	0.892788i	$-0.648746\pi$
$401$	−18.0416	−0.900956	−0.450478	+	0.892788i	$0.648746\pi$
$402$	28.9706i	1.44492i
$403$	− 27.1127i	− 1.35058i
$404$	−3.17157	−0.157792
$405$	0	0
$406$	41.7990	2.07445
$407$	− 4.34315i	− 0.215282i
$408$	30.1421i	1.49226i
$409$	−26.4853	−1.30961	−0.654806	−	0.755797i	$-0.727250\pi$
$409$	−26.4853	−1.30961	−0.654806	+	0.755797i	$0.727250\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	− 36.9706i	− 1.82141i
$413$	− 28.9706i	− 1.42555i
$414$	8.82843	0.433894
$415$	0	0
$416$	−4.10051	−0.201044
$417$	6.82843i	0.334390i
$418$	− 3.41421i	− 0.166995i
$419$	−18.8701	−0.921863	−0.460931	−	0.887436i	$-0.652485\pi$
$419$	−18.8701	−0.921863	−0.460931	+	0.887436i	$0.652485\pi$
$420$	0	0
$421$	−37.3137	−1.81856	−0.909279	−	0.416186i	$-0.863366\pi$
$421$	−37.3137	−1.81856	−0.909279	+	0.416186i	$0.863366\pi$
$422$	− 17.6569i	− 0.859522i
$423$	11.6569i	0.566776i
$424$	−17.6569	−0.857493
$425$	0	0
$426$	−30.1421	−1.46039
$427$	19.3137i	0.934656i
$428$	30.6274i	1.48043i
$429$	−3.65685	−0.176555
$430$	0	0
$431$	20.4853	0.986741	0.493371	−	0.869819i	$-0.335765\pi$
$431$	20.4853	0.986741	0.493371	+	0.869819i	$0.335765\pi$
$432$	3.00000i	0.144338i
$433$	15.0711i	0.724269i	0.932126	+	0.362135i	$0.117952\pi$
$433$	15.0711i	0.724269i	−0.932126	+	0.362135i	$0.882048\pi$
$434$	86.4264	4.14860
$435$	0	0
$436$	33.7990	1.61868
$437$	3.65685i	0.174931i
$438$	4.82843i	0.230711i
$439$	32.9706	1.57360	0.786800	−	0.617209i	$-0.211737\pi$
$439$	32.9706	1.57360	0.786800	+	0.617209i	$0.211737\pi$
$440$	0	0
$441$	4.65685	0.221755
$442$	42.6274i	2.02758i
$443$	21.3137i	1.01264i	0.862344	+	0.506322i	$0.168995\pi$
$443$	21.3137i	1.01264i	−0.862344	+	0.506322i	$0.831005\pi$
$444$	−11.7574	−0.557980
$445$	0	0
$446$	42.6274	2.01847
$447$	7.65685i	0.362157i
$448$	− 33.5563i	− 1.58539i
$449$	11.8995	0.561572	0.280786	−	0.959770i	$-0.409405\pi$
$449$	11.8995	0.561572	0.280786	+	0.959770i	$0.409405\pi$
$450$	0	0
$451$	6.48528	0.305380
$452$	17.1716i	0.807683i
$453$	21.7990i	1.02421i
$454$	−36.1421	−1.69623
$455$	0	0
$456$	−4.41421	−0.206714
$457$	− 23.1716i	− 1.08392i	−0.840404	−	0.541960i	$-0.817682\pi$
$457$	− 23.1716i	− 1.08392i	0.840404	−	0.541960i	$-0.182318\pi$
$458$	23.3137i	1.08938i
$459$	−6.82843	−0.318724
$460$	0	0
$461$	33.3137	1.55157	0.775787	−	0.630995i	$-0.217353\pi$
$461$	33.3137	1.55157	0.775787	+	0.630995i	$0.217353\pi$
$462$	− 11.6569i	− 0.542326i
$463$	− 27.8995i	− 1.29660i	−0.761385	−	0.648300i	$-0.775480\pi$
$463$	− 27.8995i	− 1.29660i	0.761385	−	0.648300i	$-0.224520\pi$
$464$	−15.2132	−0.706255
$465$	0	0
$466$	−47.4558	−2.19835
$467$	− 27.6569i	− 1.27981i	−0.768455	−	0.639903i	$-0.778974\pi$
$467$	− 27.6569i	− 1.27981i	0.768455	−	0.639903i	$-0.221026\pi$
$468$	9.89949i	0.457604i
$469$	−40.9706	−1.89184
$470$	0	0
$471$	−17.7990	−0.820134
$472$	37.4558i	1.72404i
$473$	− 4.82843i	− 0.222011i
$474$	27.3137	1.25456
$475$	0	0
$476$	−89.2548	−4.09099
$477$	− 4.00000i	− 0.183147i
$478$	13.0711i	0.597857i
$479$	−38.1838	−1.74466	−0.872330	−	0.488917i	$-0.837392\pi$
$479$	−38.1838	−1.74466	−0.872330	+	0.488917i	$0.837392\pi$
$480$	0	0
$481$	−7.94113	−0.362084
$482$	− 45.7990i	− 2.08609i
$483$	12.4853i	0.568100i
$484$	34.4558	1.56617
$485$	0	0
$486$	−2.41421	−0.109511
$487$	− 2.82843i	− 0.128168i	−0.997944	−	0.0640841i	$-0.979587\pi$
$487$	− 2.82843i	− 0.128168i	0.997944	−	0.0640841i	$-0.0204126\pi$
$488$	− 24.9706i	− 1.13036i
$489$	−11.8995	−0.538114
$490$	0	0
$491$	2.10051	0.0947945	0.0473972	−	0.998876i	$-0.484907\pi$
$491$	2.10051	0.0947945	0.0473972	+	0.998876i	$0.484907\pi$
$492$	− 17.5563i	− 0.791501i
$493$	− 34.6274i	− 1.55954i
$494$	−6.24264	−0.280870
$495$	0	0
$496$	−31.4558	−1.41241
$497$	− 42.6274i	− 1.91210i
$498$	− 15.6569i	− 0.701600i
$499$	−15.7990	−0.707260	−0.353630	−	0.935385i	$-0.615053\pi$
$499$	−15.7990	−0.707260	−0.353630	+	0.935385i	$0.615053\pi$
$500$	0	0
$501$	−10.0000	−0.446767
$502$	− 65.3553i	− 2.91695i
$503$	− 4.82843i	− 0.215289i	−0.994189	−	0.107644i	$-0.965669\pi$
$503$	− 4.82843i	− 0.215289i	0.994189	−	0.107644i	$-0.0343308\pi$
$504$	−15.0711	−0.671319
$505$	0	0
$506$	12.4853	0.555038
$507$	− 6.31371i	− 0.280402i
$508$	30.6274i	1.35887i
$509$	4.38478	0.194352	0.0971759	−	0.995267i	$-0.469019\pi$
$509$	4.38478	0.194352	0.0971759	+	0.995267i	$0.469019\pi$
$510$	0	0
$511$	−6.82843	−0.302072
$512$	31.2426i	1.38074i
$513$	− 1.00000i	− 0.0441511i
$514$	−16.4853	−0.727135
$515$	0	0
$516$	−13.0711	−0.575422
$517$	16.4853i	0.725022i
$518$	− 25.3137i	− 1.11222i
$519$	22.1421	0.971932
$520$	0	0
$521$	12.3848	0.542587	0.271293	−	0.962497i	$-0.412549\pi$
$521$	12.3848	0.542587	0.271293	+	0.962497i	$0.412549\pi$
$522$	− 12.2426i	− 0.535846i
$523$	23.7990i	1.04066i	0.853966	+	0.520329i	$0.174191\pi$
$523$	23.7990i	1.04066i	−0.853966	+	0.520329i	$0.825809\pi$
$524$	33.4142	1.45971
$525$	0	0
$526$	−9.31371	−0.406097
$527$	− 71.5980i	− 3.11886i
$528$	4.24264i	0.184637i
$529$	9.62742	0.418583
$530$	0	0
$531$	−8.48528	−0.368230
$532$	− 13.0711i	− 0.566703i
$533$	− 11.8579i	− 0.513621i
$534$	−35.5563	−1.53867
$535$	0	0
$536$	52.9706	2.28798
$537$	22.8284i	0.985119i
$538$	− 68.5269i	− 2.95440i
$539$	6.58579	0.283670
$540$	0	0
$541$	12.6274	0.542895	0.271448	−	0.962453i	$-0.412498\pi$
$541$	12.6274	0.542895	0.271448	+	0.962453i	$0.412498\pi$
$542$	60.7696i	2.61028i
$543$	24.8284i	1.06549i
$544$	−10.8284	−0.464265
$545$	0	0
$546$	−21.3137	−0.912143
$547$	5.85786i	0.250464i	0.992127	+	0.125232i	$0.0399676\pi$
$547$	5.85786i	0.250464i	−0.992127	+	0.125232i	$0.960032\pi$
$548$	− 53.5980i	− 2.28959i
$549$	5.65685	0.241429
$550$	0	0
$551$	5.07107	0.216035
$552$	− 16.1421i	− 0.687055i
$553$	38.6274i	1.64260i
$554$	55.4558	2.35609
$555$	0	0
$556$	26.1421	1.10867
$557$	− 10.0000i	− 0.423714i	−0.977301	−	0.211857i	$-0.932049\pi$
$557$	− 10.0000i	− 0.423714i	0.977301	−	0.211857i	$-0.0679510\pi$
$558$	− 25.3137i	− 1.07161i
$559$	−8.82843	−0.373403
$560$	0	0
$561$	−9.65685	−0.407713
$562$	− 25.8995i	− 1.09250i
$563$	− 14.2843i	− 0.602010i	−0.953623	−	0.301005i	$-0.902678\pi$
$563$	− 14.2843i	− 0.602010i	0.953623	−	0.301005i	$-0.0973221\pi$
$564$	44.6274	1.87915
$565$	0	0
$566$	−5.41421	−0.227576
$567$	− 3.41421i	− 0.143383i
$568$	55.1127i	2.31248i
$569$	−18.7279	−0.785115	−0.392558	−	0.919727i	$-0.628410\pi$
$569$	−18.7279	−0.785115	−0.392558	+	0.919727i	$0.628410\pi$
$570$	0	0
$571$	−19.7990	−0.828562	−0.414281	−	0.910149i	$-0.635967\pi$
$571$	−19.7990	−0.828562	−0.414281	+	0.910149i	$0.635967\pi$
$572$	14.0000i	0.585369i
$573$	17.8995i	0.747762i
$574$	37.7990	1.57770
$575$	0	0
$576$	−9.82843	−0.409518
$577$	− 1.79899i	− 0.0748929i	−0.999299	−	0.0374465i	$-0.988078\pi$
$577$	− 1.79899i	− 0.0748929i	0.999299	−	0.0374465i	$-0.0119224\pi$
$578$	71.5269i	2.97513i
$579$	−0.928932	−0.0386051
$580$	0	0
$581$	22.1421	0.918611
$582$	10.2426i	0.424571i
$583$	− 5.65685i	− 0.234283i
$584$	8.82843	0.365323
$585$	0	0
$586$	18.8284	0.777795
$587$	− 0.343146i	− 0.0141631i	−0.999975	−	0.00708157i	$-0.997746\pi$
$587$	− 0.343146i	− 0.0141631i	0.999975	−	0.00708157i	$-0.00225415\pi$
$588$	− 17.8284i	− 0.735232i
$589$	10.4853	0.432038
$590$	0	0
$591$	9.17157	0.377268
$592$	9.21320i	0.378660i
$593$	− 6.68629i	− 0.274573i	−0.990531	−	0.137287i	$-0.956162\pi$
$593$	− 6.68629i	− 0.274573i	0.990531	−	0.137287i	$-0.0438381\pi$
$594$	−3.41421	−0.140087
$595$	0	0
$596$	29.3137	1.20074
$597$	0.485281i	0.0198612i
$598$	− 22.8284i	− 0.933524i
$599$	−45.9411	−1.87710	−0.938552	−	0.345139i	$-0.887832\pi$
$599$	−45.9411	−1.87710	−0.938552	+	0.345139i	$0.887832\pi$
$600$	0	0
$601$	−23.1716	−0.945188	−0.472594	−	0.881280i	$-0.656682\pi$
$601$	−23.1716	−0.945188	−0.472594	+	0.881280i	$0.656682\pi$
$602$	− 28.1421i	− 1.14699i
$603$	12.0000i	0.488678i
$604$	83.4558	3.39577
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	− 26.1421i	− 1.06108i	−0.847661	−	0.530538i	$-0.821990\pi$
$607$	− 26.1421i	− 1.06108i	0.847661	−	0.530538i	$-0.178010\pi$
$608$	− 1.58579i	− 0.0643121i
$609$	17.3137	0.701587
$610$	0	0
$611$	30.1421	1.21942
$612$	26.1421i	1.05673i
$613$	− 37.1127i	− 1.49897i	−0.662023	−	0.749484i	$-0.730302\pi$
$613$	− 37.1127i	− 1.49897i	0.662023	−	0.749484i	$-0.269698\pi$
$614$	76.7696	3.09817
$615$	0	0
$616$	−21.3137	−0.858754
$617$	18.1421i	0.730375i	0.930934	+	0.365187i	$0.118995\pi$
$617$	18.1421i	0.730375i	−0.930934	+	0.365187i	$0.881005\pi$
$618$	− 23.3137i	− 0.937815i
$619$	4.20101	0.168853	0.0844264	−	0.996430i	$-0.473094\pi$
$619$	4.20101	0.168853	0.0844264	+	0.996430i	$0.473094\pi$
$620$	0	0
$621$	3.65685	0.146745
$622$	57.3553i	2.29974i
$623$	− 50.2843i	− 2.01460i
$624$	7.75736	0.310543
$625$	0	0
$626$	−63.9411	−2.55560
$627$	− 1.41421i	− 0.0564782i
$628$	68.1421i	2.71917i
$629$	−20.9706	−0.836151
$630$	0	0
$631$	−22.6274	−0.900783	−0.450392	−	0.892831i	$-0.648716\pi$
$631$	−22.6274	−0.900783	−0.450392	+	0.892831i	$0.648716\pi$
$632$	− 49.9411i	− 1.98655i
$633$	− 7.31371i	− 0.290694i
$634$	27.3137	1.08477
$635$	0	0
$636$	−15.3137	−0.607228
$637$	− 12.0416i	− 0.477107i
$638$	− 17.3137i	− 0.685456i
$639$	−12.4853	−0.493910
$640$	0	0
$641$	−11.4142	−0.450834	−0.225417	−	0.974262i	$-0.572375\pi$
$641$	−11.4142	−0.450834	−0.225417	+	0.974262i	$0.572375\pi$
$642$	19.3137i	0.762251i
$643$	− 42.0416i	− 1.65796i	−0.559278	−	0.828980i	$-0.688922\pi$
$643$	− 42.0416i	− 1.65796i	0.559278	−	0.828980i	$-0.311078\pi$
$644$	47.7990	1.88354
$645$	0	0
$646$	−16.4853	−0.648605
$647$	17.1127i	0.672770i	0.941725	+	0.336385i	$0.109204\pi$
$647$	17.1127i	0.672770i	−0.941725	+	0.336385i	$0.890796\pi$
$648$	4.41421i	0.173407i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	35.7990	1.40307
$652$	45.5563i	1.78412i
$653$	− 42.4264i	− 1.66027i	−0.557560	−	0.830137i	$-0.688262\pi$
$653$	− 42.4264i	− 1.66027i	0.557560	−	0.830137i	$-0.311738\pi$
$654$	21.3137	0.833432
$655$	0	0
$656$	−13.7574	−0.537135
$657$	2.00000i	0.0780274i
$658$	96.0833i	3.74572i
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	1.51472	0.0589157	0.0294579	−	0.999566i	$-0.490622\pi$
$661$	1.51472	0.0589157	0.0294579	+	0.999566i	$0.490622\pi$
$662$	− 30.9706i	− 1.20371i
$663$	17.6569i	0.685735i
$664$	−28.6274	−1.11096
$665$	0	0
$666$	−7.41421	−0.287295
$667$	18.5442i	0.718033i
$668$	38.2843i	1.48126i
$669$	17.6569	0.682653
$670$	0	0
$671$	8.00000	0.308837
$672$	− 5.41421i	− 0.208858i
$673$	2.10051i	0.0809685i	0.999180	+	0.0404843i	$0.0128901\pi$
$673$	2.10051i	0.0809685i	−0.999180	+	0.0404843i	$0.987110\pi$
$674$	−47.6985	−1.83728
$675$	0	0
$676$	−24.1716	−0.929676
$677$	11.0294i	0.423896i	0.977281	+	0.211948i	$0.0679807\pi$
$677$	11.0294i	0.423896i	−0.977281	+	0.211948i	$0.932019\pi$
$678$	10.8284i	0.415863i
$679$	−14.4853	−0.555894
$680$	0	0
$681$	−14.9706	−0.573673
$682$	− 35.7990i	− 1.37081i
$683$	− 5.65685i	− 0.216454i	−0.994126	−	0.108227i	$-0.965483\pi$
$683$	− 5.65685i	− 0.216454i	0.994126	−	0.108227i	$-0.0345173\pi$
$684$	−3.82843	−0.146384
$685$	0	0
$686$	−19.3137	−0.737401
$687$	9.65685i	0.368432i
$688$	10.2426i	0.390497i
$689$	−10.3431	−0.394042
$690$	0	0
$691$	39.1127	1.48792	0.743959	−	0.668226i	$-0.232946\pi$
$691$	39.1127	1.48792	0.743959	+	0.668226i	$0.232946\pi$
$692$	− 84.7696i	− 3.22245i
$693$	− 4.82843i	− 0.183417i
$694$	15.6569	0.594326
$695$	0	0
$696$	−22.3848	−0.848493
$697$	− 31.3137i	− 1.18609i
$698$	16.1421i	0.610989i
$699$	−19.6569	−0.743491
$700$	0	0
$701$	−11.6569	−0.440273	−0.220137	−	0.975469i	$-0.570650\pi$
$701$	−11.6569	−0.440273	−0.220137	+	0.975469i	$0.570650\pi$
$702$	6.24264i	0.235613i
$703$	− 3.07107i	− 0.115828i
$704$	−13.8995	−0.523857
$705$	0	0
$706$	−18.4853	−0.695703
$707$	− 2.82843i	− 0.106374i
$708$	32.4853i	1.22087i
$709$	12.6863	0.476444	0.238222	−	0.971211i	$-0.423435\pi$
$709$	12.6863	0.476444	0.238222	+	0.971211i	$0.423435\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	65.0122i	2.43643i
$713$	38.3431i	1.43596i
$714$	−56.2843	−2.10639
$715$	0	0
$716$	87.3970	3.26618
$717$	5.41421i	0.202198i
$718$	− 23.8995i	− 0.891921i
$719$	47.5563	1.77355	0.886776	−	0.462199i	$-0.152939\pi$
$719$	47.5563	1.77355	0.886776	+	0.462199i	$0.152939\pi$
$720$	0	0
$721$	32.9706	1.22789
$722$	− 2.41421i	− 0.0898477i
$723$	− 18.9706i	− 0.705523i
$724$	95.0538	3.53265
$725$	0	0
$726$	21.7279	0.806399
$727$	7.41421i	0.274978i	0.990503	+	0.137489i	$0.0439032\pi$
$727$	7.41421i	0.274978i	−0.990503	+	0.137489i	$0.956097\pi$
$728$	38.9706i	1.44435i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−23.3137	−0.862289
$732$	− 21.6569i	− 0.800460i
$733$	21.3137i	0.787240i	0.919273	+	0.393620i	$0.128777\pi$
$733$	21.3137i	0.787240i	−0.919273	+	0.393620i	$0.871223\pi$
$734$	40.0416	1.47796
$735$	0	0
$736$	5.79899	0.213754
$737$	16.9706i	0.625119i
$738$	− 11.0711i	− 0.407532i
$739$	−1.65685	−0.0609484	−0.0304742	−	0.999536i	$-0.509702\pi$
$739$	−1.65685	−0.0609484	−0.0304742	+	0.999536i	$0.509702\pi$
$740$	0	0
$741$	−2.58579	−0.0949912
$742$	− 32.9706i	− 1.21039i
$743$	7.31371i	0.268314i	0.990960	+	0.134157i	$0.0428326\pi$
$743$	7.31371i	0.268314i	−0.990960	+	0.134157i	$0.957167\pi$
$744$	−46.2843	−1.69686
$745$	0	0
$746$	23.8995	0.875023
$747$	− 6.48528i	− 0.237284i
$748$	36.9706i	1.35178i
$749$	−27.3137	−0.998021
$750$	0	0
$751$	25.1127	0.916375	0.458188	−	0.888855i	$-0.348499\pi$
$751$	25.1127	0.916375	0.458188	+	0.888855i	$0.348499\pi$
$752$	− 34.9706i	− 1.27525i
$753$	− 27.0711i	− 0.986525i
$754$	−31.6569	−1.15287
$755$	0	0
$756$	−13.0711	−0.475390
$757$	− 11.1716i	− 0.406038i	−0.979175	−	0.203019i	$-0.934925\pi$
$757$	− 11.1716i	− 0.406038i	0.979175	−	0.203019i	$-0.0650753\pi$
$758$	− 10.0000i	− 0.363216i
$759$	5.17157	0.187716
$760$	0	0
$761$	46.2843	1.67780	0.838902	−	0.544283i	$-0.183198\pi$
$761$	46.2843	1.67780	0.838902	+	0.544283i	$0.183198\pi$
$762$	19.3137i	0.699662i
$763$	30.1421i	1.09122i
$764$	68.5269	2.47922
$765$	0	0
$766$	28.9706	1.04675
$767$	21.9411i	0.792248i
$768$	29.9706i	1.08147i
$769$	24.3431	0.877836	0.438918	−	0.898527i	$-0.355362\pi$
$769$	24.3431	0.877836	0.438918	+	0.898527i	$0.355362\pi$
$770$	0	0
$771$	−6.82843	−0.245920
$772$	3.55635i	0.127996i
$773$	0.970563i	0.0349087i	0.999848	+	0.0174544i	$0.00555618\pi$
$773$	0.970563i	0.0349087i	−0.999848	+	0.0174544i	$0.994444\pi$
$774$	−8.24264	−0.296276
$775$	0	0
$776$	18.7279	0.672293
$777$	− 10.4853i	− 0.376157i
$778$	− 45.7990i	− 1.64197i
$779$	4.58579	0.164303
$780$	0	0
$781$	−17.6569	−0.631812
$782$	− 60.2843i	− 2.15576i
$783$	− 5.07107i	− 0.181225i
$784$	−13.9706	−0.498949
$785$	0	0
$786$	21.0711	0.751580
$787$	37.4558i	1.33516i	0.744540	+	0.667578i	$0.232669\pi$
$787$	37.4558i	1.33516i	−0.744540	+	0.667578i	$0.767331\pi$
$788$	− 35.1127i	− 1.25084i
$789$	−3.85786	−0.137344
$790$	0	0
$791$	−15.3137	−0.544493
$792$	6.24264i	0.221823i
$793$	− 14.6274i	− 0.519435i
$794$	−58.7696	−2.08565
$795$	0	0
$796$	1.85786	0.0658503
$797$	− 5.17157i	− 0.183187i	−0.995797	−	0.0915933i	$-0.970804\pi$
$797$	− 5.17157i	− 0.183187i	0.995797	−	0.0915933i	$-0.0291960\pi$
$798$	− 8.24264i	− 0.291786i
$799$	79.5980	2.81597
$800$	0	0
$801$	−14.7279	−0.520386
$802$	43.5563i	1.53803i
$803$	2.82843i	0.0998130i
$804$	45.9411	1.62022
$805$	0	0
$806$	−65.4558	−2.30558
$807$	− 28.3848i	− 0.999191i
$808$	3.65685i	0.128648i
$809$	−26.6863	−0.938240	−0.469120	−	0.883134i	$-0.655429\pi$
$809$	−26.6863	−0.938240	−0.469120	+	0.883134i	$0.655429\pi$
$810$	0	0
$811$	7.31371	0.256819	0.128410	−	0.991721i	$-0.459013\pi$
$811$	7.31371	0.256819	0.128410	+	0.991721i	$0.459013\pi$
$812$	− 66.2843i	− 2.32612i
$813$	25.1716i	0.882806i
$814$	−10.4853	−0.367509
$815$	0	0
$816$	20.4853	0.717128
$817$	− 3.41421i	− 0.119448i
$818$	63.9411i	2.23565i
$819$	−8.82843	−0.308490
$820$	0	0
$821$	0.544156	0.0189912	0.00949559	−	0.999955i	$-0.496977\pi$
$821$	0.544156	0.0189912	0.00949559	+	0.999955i	$0.496977\pi$
$822$	− 33.7990i	− 1.17888i
$823$	22.7279i	0.792246i	0.918198	+	0.396123i	$0.129645\pi$
$823$	22.7279i	0.792246i	−0.918198	+	0.396123i	$0.870355\pi$
$824$	−42.6274	−1.48500
$825$	0	0
$826$	−69.9411	−2.43356
$827$	− 3.37258i	− 0.117276i	−0.998279	−	0.0586381i	$-0.981324\pi$
$827$	− 3.37258i	− 0.117276i	0.998279	−	0.0586381i	$-0.0186758\pi$
$828$	− 14.0000i	− 0.486534i
$829$	21.5147	0.747237	0.373619	−	0.927582i	$-0.378117\pi$
$829$	21.5147	0.747237	0.373619	+	0.927582i	$0.378117\pi$
$830$	0	0
$831$	22.9706	0.796840
$832$	25.4142i	0.881079i
$833$	− 31.7990i	− 1.10177i
$834$	16.4853	0.570839
$835$	0	0
$836$	−5.41421	−0.187254
$837$	− 10.4853i	− 0.362424i
$838$	45.5563i	1.57372i
$839$	35.1127	1.21222	0.606112	−	0.795379i	$-0.292728\pi$
$839$	35.1127	1.21222	0.606112	+	0.795379i	$0.292728\pi$
$840$	0	0
$841$	−3.28427	−0.113251
$842$	90.0833i	3.10447i
$843$	− 10.7279i	− 0.369489i
$844$	−28.0000	−0.963800
$845$	0	0
$846$	28.1421	0.967547
$847$	30.7279i	1.05582i
$848$	12.0000i	0.412082i
$849$	−2.24264	−0.0769672
$850$	0	0
$851$	11.2304	0.384975
$852$	47.7990i	1.63757i
$853$	− 26.4853i	− 0.906839i	−0.891297	−	0.453419i	$-0.850204\pi$
$853$	− 26.4853i	− 0.906839i	0.891297	−	0.453419i	$-0.149796\pi$
$854$	46.6274	1.59556
$855$	0	0
$856$	35.3137	1.20700
$857$	29.9411i	1.02277i	0.859352	+	0.511385i	$0.170867\pi$
$857$	29.9411i	1.02277i	−0.859352	+	0.511385i	$0.829133\pi$
$858$	8.82843i	0.301398i
$859$	41.9411	1.43101	0.715506	−	0.698606i	$-0.246196\pi$
$859$	41.9411	1.43101	0.715506	+	0.698606i	$0.246196\pi$
$860$	0	0
$861$	15.6569	0.533584
$862$	− 49.4558i	− 1.68447i
$863$	− 8.68629i	− 0.295685i	−0.989011	−	0.147842i	$-0.952767\pi$
$863$	− 8.68629i	− 0.295685i	0.989011	−	0.147842i	$-0.0472328\pi$
$864$	−1.58579	−0.0539496
$865$	0	0
$866$	36.3848	1.23641
$867$	29.6274i	1.00620i
$868$	− 137.054i	− 4.65191i
$869$	16.0000	0.542763
$870$	0	0
$871$	31.0294	1.05139
$872$	− 38.9706i	− 1.31971i
$873$	4.24264i	0.143592i
$874$	8.82843	0.298626
$875$	0	0
$876$	7.65685	0.258701
$877$	1.89949i	0.0641414i	0.999486	+	0.0320707i	$0.0102102\pi$
$877$	1.89949i	0.0641414i	−0.999486	+	0.0320707i	$0.989790\pi$
$878$	− 79.5980i	− 2.68630i
$879$	7.79899	0.263053
$880$	0	0
$881$	3.45584	0.116430	0.0582152	−	0.998304i	$-0.481459\pi$
$881$	3.45584	0.116430	0.0582152	+	0.998304i	$0.481459\pi$
$882$	− 11.2426i	− 0.378559i
$883$	− 18.5269i	− 0.623480i	−0.950167	−	0.311740i	$-0.899088\pi$
$883$	− 18.5269i	− 0.623480i	0.950167	−	0.311740i	$-0.100912\pi$
$884$	67.5980	2.27357
$885$	0	0
$886$	51.4558	1.72869
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	13.5563i	0.454921i
$889$	−27.3137	−0.916072
$890$	0	0
$891$	−1.41421	−0.0473779
$892$	− 67.5980i	− 2.26335i
$893$	11.6569i	0.390082i
$894$	18.4853	0.618240
$895$	0	0
$896$	−70.1838	−2.34468
$897$	− 9.45584i	− 0.315721i
$898$	− 28.7279i	− 0.958663i
$899$	53.1716	1.77337
$900$	0	0
$901$	−27.3137	−0.909952
$902$	− 15.6569i	− 0.521316i
$903$	− 11.6569i	− 0.387916i
$904$	19.7990	0.658505
$905$	0	0
$906$	52.6274	1.74843
$907$	− 9.85786i	− 0.327325i	−0.986516	−	0.163663i	$-0.947669\pi$
$907$	− 9.85786i	− 0.327325i	0.986516	−	0.163663i	$-0.0523308\pi$
$908$	57.3137i	1.90202i
$909$	−0.828427	−0.0274772
$910$	0	0
$911$	−0.686292	−0.0227379	−0.0113689	−	0.999935i	$-0.503619\pi$
$911$	−0.686292	−0.0227379	−0.0113689	+	0.999935i	$0.503619\pi$
$912$	3.00000i	0.0993399i
$913$	− 9.17157i	− 0.303535i
$914$	−55.9411	−1.85037
$915$	0	0
$916$	36.9706	1.22154
$917$	29.7990i	0.984049i
$918$	16.4853i	0.544095i
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	31.7990	1.04781
$922$	− 80.4264i	− 2.64870i
$923$	32.2843i	1.06265i
$924$	−18.4853	−0.608121
$925$	0	0
$926$	−67.3553	−2.21343
$927$	− 9.65685i	− 0.317173i
$928$	− 8.04163i	− 0.263979i
$929$	0.544156	0.0178532	0.00892659	−	0.999960i	$-0.497159\pi$
$929$	0.544156	0.0178532	0.00892659	+	0.999960i	$0.497159\pi$
$930$	0	0
$931$	4.65685	0.152622
$932$	75.2548i	2.46505i
$933$	23.7574i	0.777781i
$934$	−66.7696	−2.18477
$935$	0	0
$936$	11.4142	0.373085
$937$	− 54.7696i	− 1.78924i	−0.446824	−	0.894622i	$-0.647445\pi$
$937$	− 54.7696i	− 1.78924i	0.446824	−	0.894622i	$-0.352555\pi$
$938$	98.9117i	3.22958i
$939$	−26.4853	−0.864314
$940$	0	0
$941$	13.5563	0.441924	0.220962	−	0.975282i	$-0.429080\pi$
$941$	13.5563	0.441924	0.220962	+	0.975282i	$0.429080\pi$
$942$	42.9706i	1.40006i
$943$	16.7696i	0.546092i
$944$	25.4558	0.828517
$945$	0	0
$946$	−11.6569	−0.378997
$947$	− 7.17157i	− 0.233045i	−0.993188	−	0.116522i	$-0.962825\pi$
$947$	− 7.17157i	− 0.233045i	0.993188	−	0.116522i	$-0.0371747\pi$
$948$	− 43.3137i	− 1.40676i
$949$	5.17157	0.167876
$950$	0	0
$951$	11.3137	0.366872
$952$	102.912i	3.33539i
$953$	− 34.1421i	− 1.10597i	−0.833190	−	0.552986i	$-0.813488\pi$
$953$	− 34.1421i	− 1.10597i	0.833190	−	0.552986i	$-0.186512\pi$
$954$	−9.65685	−0.312652
$955$	0	0
$956$	20.7279	0.670389
$957$	− 7.17157i	− 0.231824i
$958$	92.1838i	2.97832i
$959$	47.7990	1.54351
$960$	0	0
$961$	78.9411	2.54649
$962$	19.1716i	0.618116i
$963$	8.00000i	0.257796i
$964$	−72.6274	−2.33917
$965$	0	0
$966$	30.1421	0.969807
$967$	8.10051i	0.260495i	0.991482	+	0.130247i	$0.0415771\pi$
$967$	8.10051i	0.260495i	−0.991482	+	0.130247i	$0.958423\pi$
$968$	− 39.7279i	− 1.27690i
$969$	−6.82843	−0.219361
$970$	0	0
$971$	6.34315	0.203561	0.101781	−	0.994807i	$-0.467546\pi$
$971$	6.34315	0.203561	0.101781	+	0.994807i	$0.467546\pi$
$972$	3.82843i	0.122797i
$973$	23.3137i	0.747403i
$974$	−6.82843	−0.218797
$975$	0	0
$976$	−16.9706	−0.543214
$977$	39.5980i	1.26685i	0.773803	+	0.633426i	$0.218352\pi$
$977$	39.5980i	1.26685i	−0.773803	+	0.633426i	$0.781648\pi$
$978$	28.7279i	0.918618i
$979$	−20.8284	−0.665679
$980$	0	0
$981$	8.82843	0.281870
$982$	− 5.07107i	− 0.161824i
$983$	− 35.9411i	− 1.14634i	−0.819435	−	0.573172i	$-0.805713\pi$
$983$	− 35.9411i	− 1.14634i	0.819435	−	0.573172i	$-0.194287\pi$
$984$	−20.2426	−0.645312
$985$	0	0
$986$	−83.5980	−2.66230
$987$	39.7990i	1.26682i
$988$	9.89949i	0.314945i
$989$	12.4853	0.397009
$990$	0	0
$991$	−50.3431	−1.59920	−0.799601	−	0.600531i	$-0.794956\pi$
$991$	−50.3431	−1.59920	−0.799601	+	0.600531i	$0.794956\pi$
$992$	− 16.6274i	− 0.527921i
$993$	− 12.8284i	− 0.407098i
$994$	−102.912	−3.26416
$995$	0	0
$996$	−24.8284	−0.786719
$997$	33.5147i	1.06142i	0.847553	+	0.530711i	$0.178075\pi$
$997$	33.5147i	1.06142i	−0.847553	+	0.530711i	$0.821925\pi$
$998$	38.1421i	1.20737i
$999$	−3.07107	−0.0971643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.j.799.1		4
5.2	odd	4		285.2.a.f.1.2	✓	2
5.3	odd	4		1425.2.a.l.1.1		2
5.4	even	2	inner	1425.2.c.j.799.4		4
15.2	even	4		855.2.a.e.1.1		2
15.8	even	4		4275.2.a.x.1.2		2
20.7	even	4		4560.2.a.bj.1.1		2
95.37	even	4		5415.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.f.1.2	✓	2	5.2	odd	4
855.2.a.e.1.1		2	15.2	even	4
1425.2.a.l.1.1		2	5.3	odd	4
1425.2.c.j.799.1		4	1.1	even	1	trivial
1425.2.c.j.799.4		4	5.4	even	2	inner
4275.2.a.x.1.2		2	15.8	even	4
4560.2.a.bj.1.1		2	20.7	even	4
5415.2.a.p.1.1		2	95.37	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -3.41421i q^{7} +4.41421i q^{8} -1.00000 q^{9} -1.41421 q^{11} +3.82843i q^{12} +2.58579i q^{13} -8.24264 q^{14} +3.00000 q^{16} +6.82843i q^{17} +2.41421i q^{18} -1.00000 q^{19} -3.41421 q^{21} +3.41421i q^{22} -3.65685i q^{23} +4.41421 q^{24} +6.24264 q^{26} +1.00000i q^{27} +13.0711i q^{28} -5.07107 q^{29} -10.4853 q^{31} +1.58579i q^{32} +1.41421i q^{33} +16.4853 q^{34} +3.82843 q^{36} +3.07107i q^{37} +2.41421i q^{38} +2.58579 q^{39} -4.58579 q^{41} +8.24264i q^{42} +3.41421i q^{43} +5.41421 q^{44} -8.82843 q^{46} -11.6569i q^{47} -3.00000i q^{48} -4.65685 q^{49} +6.82843 q^{51} -9.89949i q^{52} +4.00000i q^{53} +2.41421 q^{54} +15.0711 q^{56} +1.00000i q^{57} +12.2426i q^{58} +8.48528 q^{59} -5.65685 q^{61} +25.3137i q^{62} +3.41421i q^{63} +9.82843 q^{64} +3.41421 q^{66} -12.0000i q^{67} -26.1421i q^{68} -3.65685 q^{69} +12.4853 q^{71} -4.41421i q^{72} -2.00000i q^{73} +7.41421 q^{74} +3.82843 q^{76} +4.82843i q^{77} -6.24264i q^{78} -11.3137 q^{79} +1.00000 q^{81} +11.0711i q^{82} +6.48528i q^{83} +13.0711 q^{84} +8.24264 q^{86} +5.07107i q^{87} -6.24264i q^{88} +14.7279 q^{89} +8.82843 q^{91} +14.0000i q^{92} +10.4853i q^{93} -28.1421 q^{94} +1.58579 q^{96} -4.24264i q^{97} +11.2426i q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 16 q^{14} + 12 q^{16} - 4 q^{19} - 8 q^{21} + 12 q^{24} + 8 q^{26} + 8 q^{29} - 8 q^{31} + 32 q^{34} + 4 q^{36} + 16 q^{39} - 24 q^{41} + 16 q^{44} - 24 q^{46} + 4 q^{49}+ \cdots + 12 q^{96}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 16 * q^14 + 12 * q^16 - 4 * q^19 - 8 * q^21 + 12 * q^24 + 8 * q^26 + 8 * q^29 - 8 * q^31 + 32 * q^34 + 4 * q^36 + 16 * q^39 - 24 * q^41 + 16 * q^44 - 24 * q^46 + 4 * q^49 + 16 * q^51 + 4 * q^54 + 32 * q^56 + 28 * q^64 + 8 * q^66 + 8 * q^69 + 16 * q^71 + 24 * q^74 + 4 * q^76 + 4 * q^81 + 24 * q^84 + 16 * q^86 + 8 * q^89 + 24 * q^91 - 56 * q^94 + 12 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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